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NoetherNormalization :: noetherNormalization

noetherNormalization -- data for Noether normalization

Synopsis

Description

The computations performed in the routine noetherNormalization use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.
i1 : R = QQ[x_1..x_4];
i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o2 : Ideal of R
i3 : (f,J,X) = noetherNormalization I

                    2             5                             2   2      
o3 = (map(R,R,{x  + -x  + x , x , -x  + x  + x , x }), ideal (2x  + -x x  +
                1   3 2    4   1  4 1    2    3   2             1   3 1 2  
     ------------------------------------------------------------------------
               5 3     11 2 2   2   3    2       2   2     5 2          2
     x x  + 1, -x x  + --x x  + -x x  + x x x  + -x x x  + -x x x  + x x x  +
      1 4      4 1 2    6 1 2   3 1 2    1 2 3   3 1 2 3   4 1 2 4    1 2 4  
     ------------------------------------------------------------------------
     x x x x  + 1), {x , x })
      1 2 3 4         4   3

o3 : Sequence
The next example shows how when we use the lexicographical ordering, we can see the integrality of R/ f I over the polynomial ring in dim(R/I) variables:
i4 : R = QQ[x_1..x_5, MonomialOrder => Lex];
i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);

o5 : Ideal of R
i6 : (f,J,X) = noetherNormalization I

               1     1                   10               5              
o6 = (map(R,R,{-x  + -x  + x , x , 2x  + --x  + x , 3x  + -x  + x , x }),
               6 1   2 2    5   1    1    7 2    4    1   2 2    3   2   
     ------------------------------------------------------------------------
            1 2   1               3   1  3      1 2 2    1 2       1   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + -x x  +
            6 1   2 1 2    1 5    2  216 1 2   24 1 2   12 1 2 5   8 1 2  
     ------------------------------------------------------------------------
     1   2     1     2   1 4   3 3     3 2 2      3
     -x x x  + -x x x  + -x  + -x x  + -x x  + x x ), {x , x , x })
     2 1 2 5   2 1 2 5   8 2   4 2 5   2 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 96x_1x_2x_5^6-24x_2^9x_5-3x_2^9+24x_2^8x_5^2+6x_2^8x_5-16x_2^7x_
     {-9}  | 6x_1x_2^2x_5^3-48x_1x_2x_5^5+12x_1x_2x_5^4+12x_2^9-12x_2^8x_5-x_
     {-9}  | 9x_1x_2^3+72x_1x_2^2x_5^2+36x_1x_2^2x_5+1536x_1x_2x_5^5-192x_1x_
     {-3}  | x_1^2+3x_1x_2+6x_1x_5-6x_2^3                                    
     ------------------------------------------------------------------------
                                                                           
     5^3-12x_2^7x_5^2+24x_2^6x_5^3-48x_2^5x_5^4+96x_2^4x_5^5+288x_2^2x_5^6+
     2^8+8x_2^7x_5^2+4x_2^7x_5-12x_2^6x_5^2+24x_2^5x_5^3-48x_2^4x_5^4+12x_2
     2x_5^4+96x_1x_2x_5^3+36x_1x_2x_5^2-384x_2^9+384x_2^8x_5+48x_2^8-256x_2
                                                                           
     ------------------------------------------------------------------------
                                                                             
     576x_2x_5^7                                                             
     ^4x_5^3+18x_2^3x_5^3-144x_2^2x_5^5+72x_2^2x_5^4-288x_2x_5^6+72x_2x_5^5  
     ^7x_5^2-160x_2^7x_5+8x_2^7+384x_2^6x_5^2-48x_2^6x_5-12x_2^6-768x_2^5x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3+96x_2^5x_5^2+24x_2^5x_5+18x_2^5+1536x_2^4x_5^4-192x_2^4x_5^3+96x_2^4x_
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     5^2+36x_2^4x_5+27x_2^4+216x_2^3x_5^2+162x_2^3x_5+4608x_2^2x_5^5-576x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2x_5^4+720x_2^2x_5^3+324x_2^2x_5^2+9216x_2x_5^6-1152x_2x_5^5+576x_2x_5^4
                                                                             
     ------------------------------------------------------------------------
                  |
                  |
                  |
     +216x_2x_5^3 |
                  |

             5       1
o7 : Matrix R  <--- R
If noetherNormalization is unable to place the ideal into the desired position after a few tries, the following warning is given:
i8 : R = ZZ/2[a,b];
i9 : I = ideal(a^2*b+a*b^2+1);

o9 : Ideal of R
i10 : (f,J,X) = noetherNormalization I
--warning: no good linear transformation found by noetherNormalization

                                   2       2
o10 = (map(R,R,{a + b, a}), ideal(a b + a*b  + 1), {b})

o10 : Sequence
Here is an example with the option Verbose => true:
i11 : R = QQ[x_1..x_4];
i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o12 : Ideal of R
i13 : (f,J,X) = noetherNormalization(I,Verbose => true)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20

                1                   2                           3 2          
o13 = (map(R,R,{-x  + 8x  + x , x , -x  + x  + x , x }), ideal (-x  + 8x x  +
                2 1     2    4   1  3 1    2    3   2           2 1     1 2  
      -----------------------------------------------------------------------
                1 3     35 2 2       3   1 2           2     2 2          2
      x x  + 1, -x x  + --x x  + 8x x  + -x x x  + 8x x x  + -x x x  + x x x 
       1 4      3 1 2    6 1 2     1 2   2 1 2 3     1 2 3   3 1 2 4    1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         1 2 3 4         4   3

o13 : Sequence
The first number in the output above gives the number of linear transformations performed by the routine while attempting to place I into the desired position. The second number tells which BasisElementLimit was used when computing the (partial) Groebner basis. By default, noetherNormalization tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option BasisElementLimit set to predetermined values. The default values come from the following list:{5,20,40,60,80,infinity}. To set the values manually, use the option LimitList:
i14 : R = QQ[x_1..x_4]; 
i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o15 : Ideal of R
i16 : (f,J,X) = noetherNormalization(I,Verbose => true,LimitList => {5,10})
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 10

                4      7             10                            9 2  
o16 = (map(R,R,{-x  + --x  + x , x , --x  + 8x  + x , x }), ideal (-x  +
                5 1   10 2    4   1   9 1     2    3   2           5 1  
      -----------------------------------------------------------------------
       7                 8 3     323 2 2   28   3   4 2        7   2    
      --x x  + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + --x x x  +
      10 1 2    1 4      9 1 2    45 1 2    5 1 2   5 1 2 3   10 1 2 3  
      -----------------------------------------------------------------------
      10 2           2
      --x x x  + 8x x x  + x x x x  + 1), {x , x })
       9 1 2 4     1 2 4    1 2 3 4         4   3

o16 : Sequence
To limit the randomness of the coefficients, use the option RandomRange. Here is an example where the coefficients of the linear transformation are random integers from -2 to 2:
i17 : R = QQ[x_1..x_4];
i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o18 : Ideal of R
i19 : (f,J,X) = noetherNormalization(I,Verbose => true,RandomRange => 2)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                    2       
o19 = (map(R,R,{- 2x  + x  + x , x , x  + 2x  + x , x }), ideal (- x  + x x 
                    1    2    4   1   1     2    3   2              1    1 2
      -----------------------------------------------------------------------
                      3       2 2       3     2          2      2      
      + x x  + 1, - 2x x  - 3x x  + 2x x  - 2x x x  + x x x  + x x x  +
         1 4          1 2     1 2     1 2     1 2 3    1 2 3    1 2 4  
      -----------------------------------------------------------------------
          2
      2x x x  + x x x x  + 1), {x , x })
        1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :